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%I M3706 N1514 #178 Apr 10 2024 09:58:35
%S 1,1,4,108,27648,86400000,4031078400000,3319766398771200000,
%T 55696437941726556979200000,21577941222941856209168026828800000,
%U 215779412229418562091680268288000000000000000,61564384586635053951550731889313964883968000000000000000
%N Hyperfactorials: Product_{k = 1..n} k^k.
%C A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 (this sequence) gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego, eq. (6.71.7). - _Alan Sokal_, Mar 02 2012
%C a(n) = (-1)^n/det(M_n) where M_n is the n X n matrix m(i,j) = (-1)^i/i^j. - _Benoit Cloitre_, May 28 2002
%C a(n) = determinant of the n X n matrix M(n) where m(i,j) = B(n,i,j) and B(n,i,x) denote the Bernstein polynomial: B(n,i,x) = binomial(n,i)*(1-x)^(n-i)*x^i. - _Benoit Cloitre_, Feb 02 2003
%C Partial products of A000312. - _Reinhard Zumkeller_, Jul 07 2012
%C Number of trailing zeros (A246839) increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms. - _Chai Wah Wu_, Sep 03 2014
%C Also the number of minimum distinguishing labelings in the n-triangular honeycomb rook graph. - _Eric W. Weisstein_, Jul 14 2017
%C Also shows up in a term in the solution to the generalized version of Raabe's integral. - _Jibran Iqbal Shah_, Apr 24 2021
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.
%H N. J. A. Sloane, <a href="/A002109/b002109.txt">Table of n, a(n) for n = 0..36</a>
%H Christian Aebi and Grant Cairns, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.433">Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials</a>, The American Mathematical Monthly 122.5 (2015): 433-443.
%H Mohammad K. Azarian, <a href="http://ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
%H blackpenredpen, <a href="https://www.youtube.com/watch?v=UDHGI-jRwUw">What is a Hyperfactorial?</a> Youtube video (2018).
%H CreativeMathProblems, <a href="https://www.youtube.com/watch?v=9xlVAusNrG4">A beautiful integral | Raabe's integral</a>, Youtube Video (2021).
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/glshkn/glshkn.html">Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for A002109, A000178) [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010622230958/http://www.mathsoft.com/asolve/constant/glshkn/glshkn.html">Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
%H A. M. Ibrahim, <a href="http://www.nntdm.net/papers/nntdm-19/NNTDM-19-2-30_42.pdf">Extension of factorial concept to negative numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
%H Jeffrey C. Lagarias and Harsh Mehta, <a href="http://arxiv.org/abs/1409.4145">Products of binomial coefficients and unreduced Farey fractions</a>, arXiv:1409.4145 [math.NT], 2014.
%H Jean-Christophe Pain, <a href="https://arxiv.org/abs/2304.07629">Series representations for the logarithm of the Glaisher-Kinkelin constant</a>, arXiv:2304.07629 [math.NT], 2023.
%H Vignesh Raman, <a href="https://arxiv.org/abs/2012.00882">The Generalized Superfactorial, Hyperfactorial and Primorial functions</a>, arXiv:2012.00882 [math.NT], 2020.
%H J. Sondow and P. Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl., 332 (2007), 292-314; see Section 5.
%H László Tóth, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Toth/toth9.html">Weighted gcd-sum functions</a>, J. Integer Sequences, 14 (2011), Article 11.7.7.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/K-Function.html">K-Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F a(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
%F Determinant of n X n matrix m(i, j) = binomial(i*j, i). - _Benoit Cloitre_, Aug 27 2003
%F a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - _Peter Luschny_, Jun 23 2012
%F G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^k*x). - _Paul D. Hanna_, Oct 02 2013
%F a(n) = A240993(n) / A000142(n+1). - _Reinhard Zumkeller_, Aug 31 2014
%F a(n) ~ A * n^(n*(n+1)/2 + 1/12) / exp(n^2/4), where A = 1.2824271291006226368753425... is the Glaisher-Kinkelin constant (see A074962). - _Vaclav Kotesovec_, Feb 20 2015
%F a(n) = Product_{k=1..n} ff(n,k) where ff denotes the falling factorial. - _Peter Luschny_, Nov 29 2015
%F log a(n) = (1/2) n^2 log n - (1/4) n^2 + (1/2) n log n + (1/12) log n + log(A) + o(1), where log(A) = A225746 is the logarithm of Glaisher's constant. - _Charles R Greathouse IV_, Mar 27 2020
%F From _Amiram Eldar_, Apr 30 2023: (Start)
%F Sum_{n>=1} 1/a(n) = A347345.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = A347352. (End)
%F From _Andrea Pinos_, Apr 04 2024: (Start)
%F a(n) = e^(Integral_{x=1..n+1} (x - 1/2 - log(sqrt(2*Pi)) + (n+1-x)*Psi(x)) dx), where Psi(x) is the digamma function.
%F a(n) = e^(Integral_{x=1..n} (x + 1/2 - log(sqrt(2*Pi)) + log(Gamma(x+1))) dx). (End)
%p f := proc(n) local k; mul(k^k,k=1..n); end;
%p A002109 := n -> exp(Zeta(1,-1,n+1)-Zeta(1,-1));
%p seq(simplify(A002109(n)),n=0..11); # _Peter Luschny_, Jun 23 2012
%t Table[Hyperfactorial[n], {n, 0, 11}] (* _Zerinvary Lajos_, Jul 10 2009 *)
%t Hyperfactorial[Range[0, 11]] (* _Eric W. Weisstein_, Jul 14 2017 *)
%t Join[{1},FoldList[Times,#^#&/@Range[15]]] (* _Harvey P. Dale_, Nov 02 2023 *)
%o (PARI) a(n)=prod(k=2,n,k^k) \\ _Charles R Greathouse IV_, Jan 12 2012
%o (PARI) a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1,k+1,(1+j^j*x+x*O(x^n)) )), n) \\ _Paul D. Hanna_, Oct 02 2013
%o (Haskell)
%o a002109 n = a002109_list !! n
%o a002109_list = scanl1 (*) a000312_list -- _Reinhard Zumkeller_, Jul 07 2012
%o (Python)
%o A002109 = [1]
%o for n in range(1, 10):
%o A002109.append(A002109[-1]*n**n) # _Chai Wah Wu_, Sep 03 2014
%o (Sage)
%o a = lambda n: prod(falling_factorial(n,k) for k in (1..n))
%o [a(n) for n in (0..10)] # _Peter Luschny_, Nov 29 2015
%Y Cf. A000178, A000142, A000312, A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.
%Y Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
%Y Cf. A001923, A051675, A240993, A255321, A255323, A255344.
%Y Cf. A246839 (trailing 0's).
%Y Cf. A347345, A347352.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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